OP Malhotra Class 11 Maths Solutions Chapter 27 Mathematical Reasoning Exercise 27 (A)

Question 1. Open the door.
Answer: This is a command, not a statement. A statement must be something that can be truly right or truly wrong. Commands do not have a truth value, so they are not considered statements in mathematical reasoning.
In simple words: It's a command, not a true or false statement.

🎯 Exam Tip: Focus on distinguishing commands from declarative sentences; only declarative sentences can be statements.

Question 2. 5 is a prime number.
Answer: This is a statement because it can be judged as either true or false. A prime number is a whole number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Since 5 fits this definition, the statement is true.
In simple words: This sentence is a statement because it can be true or false.

🎯 Exam Tip: A mathematical statement must have a definite truth value (true or false) to be considered a statement.

Question 3. Do you like mathematics ?
Answer: This is not a statement because it is a question. A statement must declare something that can be true or false, not ask for information. Questions do not have a truth value.
In simple words: It's a question, not something that can be true or false.

🎯 Exam Tip: Questions, commands, and exclamations are not considered statements in mathematical reasoning.

Question 4. Every rectangle is a square.
Answer: This is a statement because it has a truth value; it can be judged as either true or false. Even though this specific statement is false (not every rectangle is a square, but every square is a rectangle), it still counts as a statement because its truth can be determined.
In simple words: This sentence is a statement because it can be judged as true or false.

🎯 Exam Tip: A statement doesn't have to be true; it just has to have a definite truth value (true or false).

Question 5. Today is Sunday and tomorrow is Monday.
Answer: This is a compound statement because it combines two simpler statements using the word 'and'. The two smaller statements are:
(i) Today is Sunday
(ii) Tomorrow is Monday.
Both parts must be true for the whole compound statement to be true.
In simple words: This is a compound statement made of two smaller statements joined by 'and'.

🎯 Exam Tip: Recognize linking words like 'and' or 'or' that form compound statements from simpler ones.

Question 6. May you live long!
Answer: This is not a statement because it expresses a wish or an exclamation. Statements must be declarations that can be classified as true or false, not expressions of emotion or desire. It does not have a truth value.
In simple words: It's a wish or an exclamation, not a statement that can be true or false.

🎯 Exam Tip: Wishes and exclamations, like questions and commands, do not have truth values and are not statements.

Question 7. Rekha is studying in class eleven and she has to offer 5 subjects.
Answer: This is a compound statement. It connects two simpler statements with the word 'and'. The individual statements are:
(i) Rekha is studying in class eleven.
(ii) She has to offer 5 subjects.
Compound statements are built from two or more simpler statements joined by logical connectives like 'and'.
In simple words: This is a compound statement because it joins two simpler ideas using 'and'.

🎯 Exam Tip: Clearly identify the simple statements connected by "and" or "or" to determine if a statement is compound.

Question 8. The earth revolves around the moon.
Answer: This is a simple statement because it declares a fact that can be evaluated as true or false. It does not contain any connecting words like 'and' or 'or' to combine other statements. Though factually incorrect, it's still a statement because its truth value can be determined.
In simple words: This is a simple statement because it says one thing that can be true or false.

🎯 Exam Tip: Simple statements present a single idea without combining other statements. A simple statement has no internal logical connectors.

Question 9. New Delhi is a big city and it is the capital of India.
Answer: This is a compound statement, as it links two distinct statements using the word 'and'. The component parts are:
(i) New Delhi is a big city.
(ii) It is the capital of India.
Both of these smaller statements must be true for the entire compound statement to be true. In this case, both are true.
In simple words: This is a compound statement, made of two smaller statements joined by 'and'.

🎯 Exam Tip: Watch for conjunctions like "and" or "but" which often indicate a compound statement.

Question 10. 20 is a prime number and 20 is less than 21.
Answer: This is a compound statement, formed by connecting two simple statements with 'and'. The two simpler parts are:
(i) 20 is a prime number.
(ii) 20 is less than 21.
For an 'and' statement, both parts must be true for the whole statement to be true. In this case, 20 is not a prime number, so the first component is false, making the entire compound statement false.
In simple words: This combines two statements using 'and'. The two statements are: '20 is a prime number' and '20 is less than 21'.

🎯 Exam Tip: Remember that a prime number is a whole number greater than 1 that has exactly two distinct positive divisors: 1 and itself. 20 is divisible by 2, 4, 5, 10, so it's not prime.

Question 11. 8 is a prime number.
Answer: This is a false statement. A prime number can only be divided evenly by 1 and itself, but 8 can be divided by 1, 2, 4, and 8. Since it makes a clear claim that is demonstrably false, it is still considered a statement.
In simple words: This is a statement, but it is false because 8 is not a prime number.

🎯 Exam Tip: Always double-check definitions; a prime number has exactly two distinct positive divisors.

Question 12. Every square is a rectangle.
Answer: This is a true statement. A rectangle is a four-sided shape with four right angles, and all squares fit this definition perfectly. Since it makes a claim that is true, it is a statement.
In simple words: This is a true statement because all squares are indeed rectangles.

🎯 Exam Tip: Understand the hierarchical relationship between geometric shapes. A square is a special type of rectangle, but a rectangle is not necessarily a square.

Question 13. The earth revolves around the moon.
Answer: This is a false statement. In reality, the moon revolves around the Earth, and the Earth revolves around the sun. Despite being factually incorrect, it makes a clear assertion that can be judged as true or false, so it is a statement.
In simple words: This is a statement, but it is false. The moon goes around the Earth, not the other way.

🎯 Exam Tip: Mathematical reasoning focuses on the logical structure of a statement, which determines if it has a truth value, not just its factual correctness in the real world.

Question 14. The set of whole numbers is a finite set.
Answer: This is a false statement. The set of whole numbers includes 0, 1, 2, 3, and so on, extending endlessly without a limit. Thus, it is an infinite set, not a finite one. Since it declares something that is clearly false, it is a statement.
In simple words: This is a false statement. Whole numbers go on forever, so they form an infinite set, not a finite one.

🎯 Exam Tip: Understand the difference between finite (having a countable number of elements) and infinite (having an uncountable number of elements) sets in mathematics.

Question 15. 32 is a multiple of 8.
Answer: This is a true statement. A multiple of 8 means that when you divide 32 by 8, you get a whole number answer (32 ÷ 8 = 4). Since it makes a claim that is true, it is a statement.
In simple words: This is a true statement because 32 can be divided by 8 without any remainder.

🎯 Exam Tip: Remember that a multiple of a number is what you get when you multiply that number by an integer. For example, multiples of 8 are 8, 16, 24, 32, etc.

Question 16. 3 + 4i is a complex number.
Answer: This is a true statement. A complex number is any number that can be written as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit (where \( i^2 = -1 \)). In this case, \( a=3 \) and \( b=4 \), fitting the definition perfectly. Therefore, the statement is true.
In simple words: This is a true statement because any number written as 'a + bi' is a complex number, and 3 + 4i fits this pattern.

🎯 Exam Tip: Know the standard form of a complex number \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. The 'i' stands for the square root of -1.